# Find point M (vectors)

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• Mar 28th 2007, 02:18 AM
OReilly
Find point M (vectors)
I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.

Help anyone?
• Mar 28th 2007, 04:13 AM
ThePerfectHacker
Quote:

Originally Posted by OReilly
I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.

Help anyone?

I guess that is the center (where diagnols intersect).
Since that is the only which makes sense.
Check to see if that is true.
• Mar 28th 2007, 05:08 AM
OReilly
Quote:

Originally Posted by ThePerfectHacker
I guess that is the center (where diagnols intersect).
Since that is the only which makes sense.
Check to see if that is true.

Well, since diagonals are not always equal I can't say that is the right point.

If it were equal then it could be MC=1/2AC, MA=-1/2AC, MD = 1/2BD, MB = -1/2BD and we would have two pairs of opposite vectors.
• Mar 30th 2007, 07:53 AM
earboth
Quote:

Originally Posted by OReilly
I have to solve this problem:

Find point M in any quadrilateral ABCD such that is MA+MB+MC+MD = 0

Since LaTex doesn't work, MA,MB,MC,MD and 0 are all vectors.
Help anyone?

Hello, OReilly,

I've attached an image which show you what to do in 4 steps. The steps are numbered.

1. Construct the midpoint of AD, call it P.
2. Construct the midpoint of BC, call it Q.
3. Construct the midpoint of PQ: That's M.

If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

In step #4 you see the complete result.

Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

EB
• Mar 30th 2007, 08:59 AM
OReilly
Quote:

Originally Posted by earboth
Hello, OReilly,

I've attached an image which show you what to do in 4 steps. The steps are numbered.

1. Construct the midpoint of AD, call it P.
2. Construct the midpoint of BC, call it Q.
3. Construct the midpoint of PQ: That's M.

If you add 2 vectors the sum can be represented by the diagonal of a parallelogramm. If you add MA + MD then AD must be the other diagonal of the parallelogramm. Do the same with MB + MC.

In step #4 you see the complete result.

Obviously (MA +MD) and (MB + MC) are pointing in opposite directions and have the same length. Thus the sum of (MA + MD) + (MB + MC) = 0

EB

Thanks, very nice solution.
• Mar 31st 2007, 12:58 AM
earboth
one final remark
Hello, OReilly,

if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

If
A(a1 , a2)
B(b1 , b2)
C(c1 , c2)
D(d1 , d2)

then

M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))

EB
• Mar 31st 2007, 03:50 AM
OReilly
Quote:

Originally Posted by earboth
Hello, OReilly,

if you know the coordinates of the points A to D then you can easily calculate the coordinates of M. According to my construction you have to calculate the coordinates of a midpoint between 2 midpoints.

If
A(a1 , a2)
B(b1 , b2)
C(c1 , c2)
D(d1 , d2)

then

M(¼ (a1+b1+c1+d1), ¼ (a2+b2+c2+d2))

EB

Can this point be actually found in this way:

It is the point of intersection of lines that are dividing two pairs of opposite sides of quadrilateral in half.